Before calculating the mass of the additional Higgs bosons,
first we discuss the production mechanism of the axion and the relic abundance.
The axion can be produced through the misalignment mechanism and/or thermal production in the early Universe.
Thermally produced axion with \(\mathcal {O}\) (1) keV is regarded as warm DM. Hence, it suffers from bounds on galactic-scale structure formation, i.e., Lyman-\(\alpha \)  forest
observations [1]}, [2]}, [3]}, if it saturates all components of DM.
To safely evade the constraint, we assume that the axion is primarily produced by the misalignment mechanism[4]}, [5]}, [6]}.
